High-Order Spherical Runge-Kutta Schemes Based on Spherical Linear Extrapolation

Abstract This paper introduces a class of high-order numerical schemes for solving differential equations on the unit sphere, called Spherical Runge-Kutta methods with Richardson Extrapolation (SRKRE). Traditionally, Richardson Extrapolation (RE) is applied in Cartesian spaces through simple linear combinations of numerical solutions. Our approach adapts this idea directly to spherical geometry by extending the Spherical Linear Interpolation (SLERP) operator to perform extrapolation on the sphere itself. This results in a geometry-preserving formulation of RE that guarantees all intermediate and final solutions lie exactly on the unit sphere, without the need for any additional projection. The SRKRE schemes are constructed by combining existing low-order spherical integrators with this intrinsic extrapolation to achieve higher-order accuracy. Furthermore, we introduce a generalized counterpart, SRKREg, which extends the framework by allowing different choices of reference points on the sphere. SRKREg not only underscores the geometric foundations of SRKRE but also provides valuable insight into its accuracy properties. We analyze the stability and computational complexity of these methods and demonstrate their superior performance in both accuracy and structure preservation. Numerical experiments confirm that SRKRE and SRKREg achieve the expected order of convergence while fully honoring the geometric constraints of the problem.